Example 1: Ground State

Starting with the Thomas-Fermi solution, propagate in imaginary time to reach the ground state. Propagation smooths out the sharp edges on both components’ densities.

Physical Parameters

Atom number

\(\quad N_{\rm at} = 100\)

Atomic mass, Rubidium-87

\(\quad m = 1.4442 \times 10^{-25}~[\rm kg]\)

Trap frequencies

\(\quad (\omega_x, \omega_y, \omega_z) = 2 \pi \times (50, 50, 2000)~[{\rm Hz}]\)

\(\quad (\omega_x, \omega_y, \omega_z) = \omega_x \times (1, \gamma, \eta) = (1, 1, 40)~[\omega_x]\)

Harmonic oscillator length, x-axis

\(\quad a_x = \sqrt{\hbar / m \omega_x} = 1.525~[{\mu\rm m}]\)

3D scattering length, Rubidium-87

\(\quad a = 5.313~[{\rm nm}]\)

\(\quad a_{\rm sc} = a / a_x = 0.00348~[a_x]\)

Scattering 2D scale

\(\quad g_{\rm sc}^{2\rm D} = \sqrt{8\pi\eta}~a_{\rm sc} = 0.1105~[\omega_x a_x^2]\)

Scattering coupling

\(\quad (g_{\rm uu}, g_{\rm dd}, g_{\rm ud}) = g_{\rm sc}^{2 \rm D} \times (1, 1, 1.04)~[\omega_x a_x^2]\)

Chemical potential

\(\quad \mu = \sqrt{4 N_{\rm at} a_{\rm sc} \gamma \sqrt{\eta / 2 \pi}} = 1.875~[\omega_x]\)

Thomas-Fermi radius

\(\quad R_{\rm TF} = \sqrt{2 \mu} = 1.937~[a_x]\)

Initial population fractions

\(\quad (p_0, p_1) = (0.5, 0.5)\)

Raman wavelength

\(\quad \lambda_L = 790.1~[{\rm nm}]\)

Numerical Parameters

Number of grid points

\(\quad (N_x, N_y) = (64, 64)\)

r-grid half-size

\(\quad (x^{\rm max}, y^{\rm max}) = (8, 8)~[a_x]\)

r-grid spacing

\(\quad (\Delta x, \Delta y) = (0.25, 0.25)~[a_x]\)

k-grid half-size

\(\quad (k_x^{\rm max}, k_y^{\rm max}) = \pi / (\Delta x, \Delta y)\)

\(\quad (k_x^{\rm max}, k_y^{\rm max}) = (12.566, 12.566)~[a_x^{-1}]\)

k-grid spacing

\(\quad (\Delta k_x, \Delta k_y) = \pi / (x^{\rm max}, y^{\rm max})\)

\(\quad (\Delta k_x, \Delta k_y) = (0.3927, 0.3927)~[a_x^{-1}]\)

Time scale

\(\quad \tau_0 = 1 / \omega_x = 0.00318~[{\rm s/rad}]\)

\(\quad \tau_0 = 1~[\omega_x^{-1}]\)

Time step duration, imaginary

\(\quad \Delta \tau_{\rm im} = 1 / 50~[-i \tau_0]\)

Number of time steps, imaginary

\(\quad N_{\rm im} = 100\)

import os
import sys
sys.path.insert(0, os.path.abspath('../..'))  # Adds project root to the PATH

import numpy as np

from spinor_gpe.pspinor import pspinor as spin


# 1. SETUP

DATA_PATH = 'examples/Trial_011'  # Default data path is in the /data/ folder

FREQ = 50
W = 2*np.pi*FREQ
Y_SCALE = 1
Z_SCALE = 40.0

ATOM_NUM = 1e2
OMEG = {'x': W, 'y': Y_SCALE * W, 'z': Z_SCALE * W}
G_SC = {'uu': 1, 'dd': 1, 'ud': 1.04}

ps = spin.PSpinor(DATA_PATH, overwrite=True,  # Initialize PSpinor object
                  atom_num=ATOM_NUM,
                  omeg=OMEG,
                  g_sc=G_SC,
                  pop_frac=(0.5, 0.5),
                  r_sizes=(8, 8),
                  mesh_points=(64, 64))

ps.coupling_setup(wavel=790.1e-9, kin_shift=False)

ZOOM = 4  # Zooms the momentum-space density plots by a constant factor

# Plot real- and momentum-space density & real-space phase of both components
ps.plot_spins(rscale=ps.rad_tf, kscale=ps.kL_recoil, zoom=ZOOM)


# 2. RUN (Imaginary-time)

DT = 1/50
N_STEPS = 100
DEVICE = 'cpu'
ps.rand_seed = 99999

# Run propagation loop:
# - Returns `PropResult` & `TensorPropagator` objects
res, prop = ps.imaginary(DT, N_STEPS, DEVICE, is_sampling=True, n_samples=50)


# 3. ANALYZE

res.plot_spins(rscale=ps.rad_tf, kscale=ps.kL_recoil, zoom=ZOOM)
res.plot_total(kscale=ps.kL_recoil, zoom=ZOOM)  # Plot total density & phase
res.plot_pops()  # Plot how the spins' populations evolves
res.make_movie(rscale=ps.rad_tf, kscale=ps.kL_recoil, play=True, zoom=ZOOM,
               norm_type='half')